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The Ptolemaic is based on the Greek philosophy that all the objects in the sky must follow circular motion in one form or another. With the Earth supposed at the centre, the planets' movements were quite complicated and each planet needed a good number of epicycles.

The idea of epicycles is that you have a large circle (the deferent) and then a small circle (the epicycle) whose centre moves round the deferent. Then the planet travels round that epicycle.

Extra epicycles were required to represent what we now know as the planets' elliptical orbits and the inclination of the orbits to the ecliptic.

The basic idea of the deferent and epicycle is easily explained in terms of Venus's orbit. We now know that with the Sun at the centre, Venus and the Earth travel round circular orbits in 225¾ and 365¼ days with radii of 1.000 units for the Earth and 0.723 units for Venus (we are assuming circular orbits to make the example easier). This is the basis of the Copernican heliocentric model.

If you choose to consider the Earth is at the centre, this motion is exactly replicated by having a deferent with radius of 1.000 units round the Earth, and on it is an epicycle with its centre going round in 365¼ days. The epicycle has a radius of 0.723 units and Venus travels round the epicycle in 225¾ days. This is the basis of Ptolemy's model (in the Ptolemaic model the Sun also went round the Earth in 365¼ days, but at a larger radius beyond the 'sphere' of Venus).

Both these models represent the position of Venus as seen from the Earth with equal accuracy. That is still the case when the extra epicycles are added for ellipticity, inclination et cetera.

So the issue between the Ptolemaic model (geocentric) and the Copernican model (heliocentric) was not accuracy but whether the Earth or Sun was at the centre. There was no way at that time of deciding which was correct, but Copernicus's model did not have such violent motion of Venus, which was one reason Galileo preferred it.

Eventually Kepler came up with an alternative system, and that is the one used today, because it is backed up by later theoretical discoveries of the law of gravity and the laws of motion.

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